Cauchy Integral Theorem over a square root

Question

How do I evaluate $$ \oint\limits_{|z|=1}\sqrt z\mathrm{d} z $$ using Cauchy's Integral Theorem?

Answer 1

Assuming you're using the principal branch of $f(z) = \sqrt{z}$, which has a branch cut on the negative real axis, you have $\sqrt{e^{i\theta}} = e^{i\theta/2}$ for $-\pi < \theta < \pi$, so your integral is $$ \int_{-\pi}^\pi e^{i\theta/2} i e^{i\theta} d\theta = - 4 i/3$$ To do this using Cauchy's theorem, you deform your contour to one that goes from $-1$ to $0$ just below the branch cut, and then back to $-1$ just above the branch cut, thus $$ 2 (-i) \int_{-1}^0 \sqrt{-t}\; dt = -4\pi i/3$$